Why will 'Copy Rotation' constraint make this object point in an unexpected direction?
From my experience, unless you have a good reason to use the Copy Rotation constraint, it is best to parent the object in question to the armature. If, as I believe to be the case here, there's no need to deform the mesh (i.e. all movable parts are separate objects), you can use the Bone parenting mode. Using this mode will make it so you're not required to use weight groups, which are irrelevant if the object is not to be deformed but only transformed.